Questions — CAIE P2 (709 questions)

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CAIE P2 2015 November Q5
8 marks Standard +0.3
5 It is given that \(\int _ { 0 } ^ { a } \left( 3 \mathrm { e } ^ { 3 x } + 5 \mathrm { e } ^ { x } \right) \mathrm { d } x = 100\), where \(a\) is a positive constant.
  1. Show that \(a = \frac { 1 } { 3 } \ln \left( 106 - 5 \mathrm { e } ^ { a } \right)\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
CAIE P2 2015 November Q6
9 marks Standard +0.3
6
  1. Express \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence
    1. solve the equation \(( \sqrt { } 5 ) \cos \theta - 2 \sin \theta = 0.9\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\),
    2. state the greatest and least values of $$10 + ( \sqrt { } 5 ) \cos \theta - 2 \sin \theta$$ as \(\theta\) varies.
CAIE P2 2015 November Q7
10 marks Standard +0.3
7 The equation of a curve is \(y = \frac { \sin 2 x } { \cos x + 1 }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 \left( \cos ^ { 2 } x + \cos x - 1 \right) } { \cos x + 1 }\).
  2. Find the \(x\)-coordinate of each stationary point of the curve in the interval \(- \pi < x < \pi\). Give each answer correct to 3 significant figures.
CAIE P2 2016 November Q1
3 marks Easy -1.2
1 Solve the equation \(| 0.4 x - 0.8 | = 2\).
CAIE P2 2016 November Q2
5 marks Moderate -0.3
2
  1. Given that \(\frac { 1 + 4 ^ { y } } { 3 + 2 ^ { y } } = 5\), find the value of \(2 ^ { y }\).
  2. Use logarithms to find the value of \(y\) correct to 3 significant figures.
CAIE P2 2016 November Q3
6 marks Moderate -0.8
3 The definite integral \(I\) is defined by \(I = \int _ { 0 } ^ { 2 } \left( 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3 \right) \mathrm { d } x\).
  1. Show that \(I = 8 \mathrm { e } - 2\).
  2. Sketch the curve \(y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } x } + 3\) for \(0 \leqslant x \leqslant 2\).
  3. State whether an estimate of \(I\) obtained by using the trapezium rule will be more than or less than \(8 \mathrm { e } - 2\). Justify your answer.
CAIE P2 2016 November Q4
8 marks Moderate -0.3
4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where \(a\) is a constant.
  1. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
  2. Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
  3. When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.
CAIE P2 2016 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-2_556_844_1731_648} The diagram shows the curve \(y = \frac { 4 \ln x } { x ^ { 2 } + 1 }\) and its stationary point \(M\). The \(x\)-coordinate of \(M\) is \(m\).
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(m = \mathrm { e } ^ { 0.5 \left( 1 + m ^ { - 2 } \right) }\).
  2. Use an iterative formula based on the equation in part (i) to find the value of \(m\) correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
CAIE P2 2016 November Q6
10 marks Challenging +1.2
6
  1. Show that \(\frac { \cos 2 \theta } { 1 + \cos 2 \theta } \equiv 1 - \frac { 1 } { 2 } \sec ^ { 2 } \theta\).
  2. Solve the equation \(\frac { \cos 2 \alpha } { 1 + \cos 2 \alpha } = 13 + 5 \tan \alpha\) for \(0 < \alpha < \pi\).
  3. Find the exact value of \(\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \frac { \cos x } { 1 + \cos x } \mathrm {~d} x\).
CAIE P2 2016 November Q7
10 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{9bbcee46-c5b8-4836-a4b4-f317bf8b1c0a-3_533_698_735_717} The diagram shows the curve with parametric equations $$x = 4 \sin \theta , \quad y = 1 + 3 \cos \left( \theta + \frac { 1 } { 6 } \pi \right)$$ for \(0 \leqslant \theta < 2 \pi\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed in the form \(k ( 1 + ( \sqrt { } 3 ) \tan \theta )\) where the exact value of \(k\) is to be determined.
  2. Find the equation of the normal to the curve at the point where the curve crosses the positive \(y\)-axis. Give your answer in the form \(y = m x + c\), where the constants \(m\) and \(c\) are exact.
CAIE P2 2017 November Q1
4 marks Moderate -0.3
1 Candidates answer on the Question Paper.
Additional Materials: List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} Write your Centre number, candidate number and name in the spaces at the top of this page.
Write in dark blue or black pen.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50. This document consists of 11 printed pages and 1 blank page. 1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.
CAIE P2 2017 November Q2
5 marks Standard +0.3
2 Solve the equation \(5 \cos \theta ( 1 + \cos 2 \theta ) = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2017 November Q3
7 marks Standard +0.3
3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.
CAIE P2 2017 November Q4
8 marks Standard +0.3
4
  1. Find \(\int \frac { 4 + \sin ^ { 2 } \theta } { 1 - \sin ^ { 2 } \theta } \mathrm {~d} \theta\).
  2. Given that \(\int _ { 0 } ^ { a } \frac { 2 } { 3 x + 1 } \mathrm {~d} x = \ln 16\), find the value of the positive constant \(a\).
CAIE P2 2017 November Q5
8 marks Moderate -0.3
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } + 37 x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\). It is also given that the remainder is 40 when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P2 2017 November Q6
9 marks Standard +0.3
6 The parametric equations of a curve are $$x = 2 \mathrm { e } ^ { 2 t } + 4 \mathrm { e } ^ { t } , \quad y = 5 t \mathrm { e } ^ { 2 t }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence find the coordinates of the stationary point, giving each coordinate correct to 2 decimal places.
  2. Find the gradient of the normal to the curve at the point where the curve crosses the \(x\)-axis.
CAIE P2 2017 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ebfaafca-3279-44f8-a910-0756ffa25c70-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).
  1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
  2. Show by calculation that \(- 4.5 < q < - 4.0\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
CAIE P2 2017 November Q1
4 marks Moderate -0.8
1 Use logarithms to solve the equation \(5 ^ { 3 x - 1 } = 2 ^ { 4 x }\), giving your answer correct to 3 significant figures.
CAIE P2 2017 November Q2
4 marks Standard +0.3
2 It is given that \(x\) satisfies the equation \(| x + 1 | = 4\). Find the possible values of $$| x + 4 | - | x - 4 | .$$
CAIE P2 2017 November Q3
6 marks Standard +0.3
3 The equation of a curve is \(y = \tan \frac { 1 } { 2 } x + 3 \sin \frac { 1 } { 2 } x\). The curve has a stationary point \(M\) in the interval \(\pi < x < 2 \pi\). Find the coordinates of \(M\), giving each coordinate correct to 3 significant figures.
CAIE P2 2017 November Q4
7 marks Moderate -0.3
4 The polynomials \(\mathrm { p } ( x )\) and \(\mathrm { q } ( x )\) are defined by $$\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } + a x - 15 \quad \text { and } \quad \mathrm { q } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + b x + 21 ,$$ where \(a\) and \(b\) are constants. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\) and also of \(\mathrm { q } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. Show that the equation \(\mathrm { q } ( x ) - \mathrm { p } ( x ) = 0\) has only one real root. \includegraphics[max width=\textwidth, alt={}, center]{e2b16207-2cb7-412b-ba7f-758e4d3f1ffb-06_631_643_260_749} The diagram shows the curve \(y = 4 e ^ { - 2 x }\) and a straight line. The curve crosses the \(y\)-axis at the point \(P\). The straight line crosses the \(y\)-axis at the point \(( 0,9 )\) and its gradient is equal to the gradient of the curve at \(P\). The straight line meets the curve at two points, one of which is \(Q\) as shown.
CAIE P2 2017 November Q6
10 marks Moderate -0.3
6
  1. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin x ( 4 \sin x + 6 \cos x ) \mathrm { d } x\).
  2. Given that \(\int _ { 0 } ^ { a } \frac { 6 } { 3 x + 2 } \mathrm {~d} x = \ln 49\), find the value of the positive constant \(a\).
CAIE P2 2017 November Q7
10 marks Standard +0.3
7 The equation of a curve is \(x ^ { 2 } + 4 x y + 2 y ^ { 2 } = 7\).
  1. Find the equation of the tangent to the curve at the point \(( - 1,3 )\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
  2. Show that there is no point on the curve at which the gradient is \(\frac { 1 } { 2 }\).
CAIE P2 2017 November Q1
4 marks Moderate -0.3
1 Solve the equation \(\ln ( 3 x + 1 ) - \ln ( x + 2 ) = 1\), giving your answer in terms of e.
CAIE P2 2017 November Q7
9 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_542_789_260_676} The diagram shows the curve $$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$ The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).
  1. Find the value of \(m\) and hence show that \(q\) satisfies the equation $$x = a \sin \frac { 1 } { 2 } x + b ,$$ where the values of the constants \(a\) and \(b\) are to be determined.
  2. Show by calculation that \(- 4.5 < q < - 4.0\).
  3. Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.